Modal Logic for Reasoning About Uncertainty and Confusion

Abstract

We consider a modal logic that can formalise statements about uncertainty and beliefs such as `I think that my wallet is in the drawer rather than elsewhere' or `I am confused whether my appointment is on Monday or Tuesday'. To do that, we expand G\"odel modal logic KG with the involutive negation ~ defined as v(~A,w)=1-v(A,w). We provide semantics with the finite model property for our new logic that we call KG_inv and show its equivalence to the standard semantics over [0,1]-valued Kripke models. Namely, we show that a formula is valid in the standard semantics of KG_inv iff it is valid in the new semantics. Using this new semantics, we construct a constraint tableaux calculus for KG_inv that allows for an explicit extraction of countermodels from complete open branches and then employ the tableaux calculus to obtain the PSPACE-completeness of the validity in KG_inv.